### conditions (after Jefferies and Shuttle, 2002, with permission Institution of Civil Engineers).

Method (c) is to shear the soil to large displacement in a ring shear device and measure the limiting stress conditions. Negussey et al. (1988) describe a series of

experiments they performed with a ring shear device on Ottawa sand (a quartz sand), two
tailings sands, granular copper, lead shot and glass beads. These tests showed how the
large strain for each material is invariant with normal stress. The great difficulty with
ring shear, however, is that the complete stress conditions are not known and only a
*friction angle can be determined (i.e. p′ is simply unknown in ring shear). The method *
also relies on the tacit assumption that is invariant with the proportion of intermediate
principal stress. This is known to be incorrect.

*The best approach to determining M**tc* is to use a mixture of Method (b) and Method
(a). The testing to determine the CSL will provide data that should be reduced as per
Method (b), as it is in effect free information. Then a few drained triaxial compression
*tests should be carried out on at least one dense (say ψ<−0.2) and one compact (ψ≈−0.1) *
sample. These tests allow a plot as per Method (a), albeit with limited data points. The
dense sample data can then be reduced to stress-dilatancy form. Iterating between the
*forms of data presentation will allow reasonably constrained estimates of M**tc* (within
about ±0.03) within normal budgets of geotechnical projects. For large projects or
research programs, it is best to carry out the detailed testing of Method (a).

**2.4.4 Stress dilatancy and critical friction ratio **

*Rowe’s stress dilatancy relationship is given as equation [2.2] where the parameter K is *
related to the mineral to mineral friction Rowe suggested that the operating sliding
contact friction angle in [2.2] was not and was such that (where
is the critical state angle). Rowe’s suggestion has not been disputed and appears in
current geotechnical textbooks (e.g. Wood 1990). The reason the operating friction is
*required to change is that the coefficient K in equation [2.2] is used to relate strain rates *
to shear stress levels and is not just the stress ratio at the peak or critical state. Minimal
guidance is provided in the literature on how *evolves, and K is generally taken as *
constant.

*The stress-dilatancy behaviour of Erksak sand is presented in η–D** ^{P}* space on Figure
2.27. Dense and loose sand data are shown separately for clarity in Figures 2.27a and b.

*Stress-dilatancy, for both loose and dense samples, is approximately linear in the η -D** ^{P}*
space between the low stress ratio initial part of the curve and the peak stress ratio. In the
case of the dense sand, the stress-dilatancy plots naturally reverse once the stress ratio
reaches a peak, giving a “hook” in the curve as it drops to the critical state. The initial
non-linear part of the curve at low stress ratio is not considered representative of stress
dilatancy and is attributed to the effects of initial fabric and apparent overconsolidation.

The best fit linear trends through the loose sand data (Figure 2.27b) project to the
*D*^{P}*=0 axis at a value of η that suggests M≈1.22 for ψ>0. This projection is based on the *
*trends seen in the data in the range 0.1<D*^{P}*<0.6 so as to avoid the effects of initial *
*elasticity and inaccuracy at large displacements. The projected value is M (rather than *
*M**f**), as the D** ^{P}*=0 situation is the critical state for the loose samples. The fit of one straight

*line to the data suggests that M*

*f*

*≈M for most of the stress path.*

The dense sands, Figure 2.27a, show a different behaviour from loose sands. A best
*estimate is about M**f**≈1.05 based on the pre-peak linear part of the stress-dilatancy plot for *

Dilatancy and the state parameter 85

*ψ<0, where M**f** corresponds to where the straight line crosses the D** ^{P}*=0 axis. This is
obviously not the critical state because dilation continues into negative dilatancy space.

*This transient condition where D** ^{P}*=0, i.e. where volumetric strain rate changes from

*contraction to dilation, is called the image condition here. Ishihara (1975) called this the*phase transformation while others have called it the pseudo steady state. The term image

*condition is preferred as it is a projection of the critical stress ratio on the q–p plane,*whereas it is neither a phase change nor a steady state nor a pseudo condition.

There are now two possible equations for the general stress dilatancy function
expressed in equation [2.5]. The first is suggested by Figure 2.26, assuming the slope of
*the fitted trend line is given by (being careful about the negative sign of D**min**): *

[2.12]

*The value of N scales how the friction ratio varies from the critical value for a given *
*dilatancy, so that M is in fact a constant M*tc. The second plausible function is based on
Figure 2.27 and expressed as

[213]

*In [2.13] the stress ratio at the image condition M**i** varies, and the scaling factor N in *
*[2.12] is in effect zero. Having N=0 is simpler, but it is necessary to define how M**i*

evolves or changes during shearing.

*There are only two material state variables to control the evolution of M**i**; the state *
*parameter ψ and a tensor describing the arrangement of particle contacts (“fabric”). A *
*combination of state parameter and fabric would also be acceptable. Because ψ is the *
only state variable that can be reasonably determined with current technology, Figure
*2.28 shows how M**i** varies with ψ for Erksak sand. For dense sand, the filled squares *
*shown on Figure 2.28 are M**i* values obtained from where the stress-dilatancy paths cross
*the D*^{P}*=0 axis on Figure 2.27. In the case of the loose sands M**i* cannot be obtained by
inspection although the data on Figure 2.27 show nearly straight line trends once shear